googologywikiaorg-20200223-history
User blog:Ecl1psed276/Introduction and Analysis of Star Notation - Part 1
Hey everyone, I'd like to introduce my new notation called Star Notation. Basics The first rule of Star Notation is this: a0b = a+b. So 305 = 8, and 2760276 = 552. Then, we say that a00b = a0a0a0a....a0a (where there are b a's). Expressions like this are evaluted from right to left, so for example 40506 = 40(506). It is fairly easy to see that a00b is equal to a*b, becuase we are just adding a to itself b times. The next step is to say a000b = a00a00a00a...a00a (where there are b a's). a000b is equivalent to exponentiation. Then of course we can say a0000b = a000a000a...a000a (where there are b a's), and we can continue adding 0's for as long as we want. So here is what we have so far: a0b is addition, which has level 0 in the FGH. a00b is multiplication, which has level 1. a000b has level 2 (exponentiation), a0000b has level 3 (tetration), etc. So the ordinal limit of our notation so far is \(\omega\). Bracket Notation From now on, we will compare the ordinal level of the operators, instead of the entire expression. So in a000b (for example) we will only consider the 000 part, and we will ignore the a and b. This is because the basic rules will stay the same throughout the entire notation. In my notation, the limit of 000....00 is 1. So 1 has level \(\omega\) in the FGH. The next operator is 10, so a10b expands to a1a1...a1a with b a's. Then we can have 100, 1000, etc. Then we have 11, which is the limit of 1000...0, and 111, 1111, and so on. Then we reach 2, which is the limit of 11...1. We can also go to 3, 4, and so on. At this point, we need to make a generalization. Whenever we see a 1, we replace it with 0. So 1 is just a shorthand for 0. And 2 is just a shorthand for 00, 3 is just a shorthand for 000, etc... Now we have a fairly obvious way to go further: Since the limit of 000...0 is 1, the limit of [0000...00] should be 1 as well. Then, we can have things like 10, 11, 11, and so on. The limit of the sequence 111...1 is [10]. Basically, whenever we see a 0 at the end of a bracket, we remove it and copy the bracket b times. So [X0] is the limit of XXX...XX for all X. Here is the ruleset for Bracket Notation: #a0b = a+b. End process. #aX0b = aXaXaXa...aXa (with b instances of a). End process. #Otherwise, the expression is of the form aXb. Set the "current expression" to be X. #Let the current expression be \(X_1X_2...X_n\) for some \(X_1, X_2,...X_n.\) #If \(X_n\) is of the form Y0 for some Y, replace \(X_n\) with YY...Y with b instances of Y. End process. #Otherwise, set the current expression to be \(X_n\), and go back to rule 4. Analysis 2 = [00] reaches the limit of Conway Chained Arrow Notation, 4-entry BEAF/BAN arrays, and the ### delimiter in ExE. 1 = [0] reaches the limit of linear BEAF/BAN arrays, and the limit of Extended Hyper-E Notation. [11] reaches the limit of 2-row BEAF arrays. 2 reaches the limit of planar BEAF arrays. [1] reaches the limit of dimensional BEAF arrays. [[1]] reaches the limit of hyperdimensional BEAF arrays. Finally, the limit ordinal of Bracket Notation is \(\varepsilon_0\), which is the limit of tetrational BEAF arrays and Cascading-E notation. Linear Array Notation Now we will introduce a new mechanic to the notation: We can put an array inside a pair of brackets. For example, [1,4,1,10] is a valid array. Arrays are evaluated like this: *First, find the leftmost nonzero element in the array. *If this element does not end in 0, we jump into the element and decompse that. *If this element ends in 0, and it is the leftmost element in the array, we expand the array in much the same way as Bracket Notation. For example, [20,5,1] expands to [2,5,1][2,5,1][2,5,1]...[2,5,1]. *We must introduce a new rule for if this element ends in 0 and it is not the leftmost element. The rule is this: 0,0...0,0,a+1,b...y,z will expand to [0,0...0,[0,0...0,[0,0...0,...,a,b...y,z],a,b...y,z],a,b...y,z]. Alternatively, 0,0...0,0,a+1,b...y,z represents the fixed point of X -> 0,0...0,X,a,b...y,z, much like the Veblen Phi function. Here is the complete ruelset for Linear Array Notation: #a0b = a+b. End process. #aX0b = aXaXaXa...aXa (with b instances of a). End process. #Otherwise, the expression is of the form aXb. Set the "current expression" to be X. #Let the current expression be \(X_1X_2...X_n\) for some \(X_1, X_2,...X_n.\) #Find the first nonzero element in the array \(X_n\), and call it Z. #If Z is the first element, and Z is of the form Y0 for some Y, replace \(X_n\) with \(X*X*...X*\) with b instances of \(X*\). In this case, \(X*\) is the same as \(X_n\), but with the first nonzero element replaced by Y. End process. #If Z is not the first element and it is of the form Y0, do this subroutine: ##Set K to be 0. ##Make an array that is the same as \(X_n\), but with the 0 stripped off from Z. ##Find the previous element in the array (which is a 0), and replace it with K. ##Set this array to be the new value of K. ##Repeat steps 2-4 exactly b times. ##Replace \(X_n\) with K in the main array. End process. #If Z is not of the form Y0, set the current expression to be \(X_n\), and go back to rule 4. Using these rules, the limit of [[[[..[[0]]..]]] is 0,1. Then, 1,1 is the limit of 0,10,1...0,1. [0,1,1] is the limit of [[[..[[0..]]],1]. 0,2 is the limit of [[[[...[[0,1],1],1]...,1],1]. 0,0,1 is the limit of [0,[0,[0...[0,0]]..]]], and so on. The limit of Linear Array Notation is 0,0,0,0....0,0,1 with arbitrarily many entries. Analysis So the limit of Linear Array Notation is \(\varphi(\omega,0)\), which is also the limit of 3-entry array-arrays in BEAF, linear arrays in HAN, and the #^^#^# delimiter in ExE. Alpha Dimensional Array Notation (ADAN) In ADAN, we will introduce the idea of separators. The smallest separator is the comma, which is a shorthand for (0). The next separator is (0)(0), just as the next operator after 0 is 00. Then we have (0)(0)(0), (0)(0)(0)(0), and so on. The limit of this sequence is (1), or (0). But how exactly do the separators work? When we find the first nonzero element in the array, we have to check the separator immediately before it. If it is a comma, or (0), we use the previous rules to expand the array. If the seperator ends in (0), for example (0,10)(0), then we will expand the array like this: X(0) b... = X 0 X 0 X ... X 0 X 1 X(0) b-1... This rule might be hard to understand, so I will give a few examples. [0(0,10)(0)3] expands to [0(0,10)0(0,10)0...0(0,10)1(0,10)(0)2] [0(1)(0)10] expands to [0(1)0(1)0(1)0...0(1)0(1)(0)1]. (Remember, in this case, the array has 2 entries, 0 and 10. And it has only a single seperator, (1)(0).) So the limit of (0)(0)(0)...(0)(0) is (1), the limit of (1)(1)...(1) is (2), and so on. We can even do things like (1), (0,1), (0(1)1), etc. The limit of ADAN is the fixed point of a -> 0(a)1. Analysis So the limit of Alpha Dimensional Array Notation is \(\Gamma_0\), which is the limit of expandal arrays in BEAF, and the level of the #^^^# seperator in ExE. Beta Dimensional Array Notation (BDAN) The next step in continuing dimensional array notation is to allow seperators within seperators. So we can have things like 0(0(3)2)1. The smallest array with a seperator inside a seperator is 0(0,1)1. Since the comma takes a fixed point, this is the fixed point of a -> 0(a)1 which we know from the previous section to be \(\Gamma_0\). That was a pretty short explanation, but everything you need to know was already covered in the previous section. Let's jump right into the analysis. Analysis 0(0,1)(0)1 reaches the limit of Extended Cascading-E and 4-entry array-arrays in BEAF. 0(0,1)(0,1)1 reaches the limit of Hyper-Extended Cascading-E and {X,X,1,1,2} arrays in BEAF. 0(1,1)1 reaches the limit of linear array-arrays in BEAF. 0(0,2)1 reaches the limit of {X,X,2(1)2} arrays in BEAF, and dimensional arrays in HAN. [0(0,1)1] reaches the limit of dimensional array-arrays in BEAF. The limit of BDAN is \(\psi(\varepsilon_{\Omega+1})\), which is the limit of tetrational array-arrays in BEAF, and the growth rate of Bird's H(n) function. Alpha Needanameforthis Notation TODO!!! Analysis So the limit of 0,,(0,,(0...(0,,(0,,1)1)1...1)1)1 is \(\psi(\varepsilon_{\Omega_2+1}). This is also equivalent to 0,,,1. Category:Blog posts